How to Normalize a Vector

A vector is an object that has both magnitude (sometimes called size or length) and direction. Vectors are usually represented by drawing an arrow, where the direction of the arrow represents the direction of the vector, and the length of the arrow represents its magnitude. Normalizing a vector involves converting it to a “unit vector” with a standard magnitude, usually 1, while preserving the vector’s original direction. To do this, start by determining the start and end points of your vector. For instance, the vector may start at (0,0) on the x-y axis, and end at (3,4). This vector moves up from left to right. From there, you can determine that your unit vector will have the same starting point and direction as the original vector. You also know that the length of your unit vector is 1. Now you’ll need to calculate the end point, or terminal point, of your unit vector. First, calculate the length of the original vector using the Pythagorean theorem, a^2 + b^2 = c^2. Think of the vector as a right triangle, where sides A and B equal the values of the end coordinates in the x and y axes, and the hypotenuse is the length of the vector. In this case, we know that 32 + 42 = 25. Take the square root of 25 to get 5, the length of the vector. Finally, divide the x and y coordinates by the length of the vector to get the endpoint coordinates of your normalized vector. Now you know that the start point of your unit vector is (0,0), its end point is (3/5, 4/5), its magnitude is 1, and it moves up as you go from left to right along the x axis. To learn how to normalize a vector in 2-dimensional or n-dimensional space, keep reading!